TITLE: Examples of Morita equivalent rings
QUESTION [9 upvotes]: Can someone give some examples of Morita equivalent rings different from the classical one? (i.e. that a ring $R$ is Morita equivalent to the ring $M_n(R)$)

REPLY [4 votes]: Consider the algebra of upper triangular $2 \times 2$ matrices$$\left( \begin{array}{cc} * & * \\ 0 & * \end{array} \right)$$
As a module over itself it is sum of two indecomposable modules. Denote them as $V_1$ and $V_2$ (one and two dimensional respectively). $P = V_1^{\oplus n} \oplus V_2 ^ {\oplus m}$ is a projective generator (here $n,m \geq 1$). For instance, n=2, m=1 provides $End(P)$ is isomorphic to
$$ \left( \begin{array}{ccc} * & * & 0 \\ * & * & 0 \\ * & * & * \end{array} \right)$$